## Non-existence of positive solutions to semilinear elliptic equations on $$\mathbb{R}^n$$ or $$\mathbb{R}_+^n$$ through the method of moving planes.(English)Zbl 0910.35048

The paper is mainly concerned with the semilinear elliptic problem $\Delta u+K \bigl(| x| \bigr)u^p=0 \quad \text{in } \mathbb{R}^n\;(n\geq 3), \quad u>0,$ where $$K(| x|)$$ is nonnegative. The non-existence of a radially symmetric solution of (*) has been shown for $$p= (n+2)/(n-2)$$, and for $$p>1$$ under suitable monotonicity conditions on $$K(r)$$. Here is proved the non-existence of a non-radial solution for the case where either $$p>(n+2)/(n-2)$$ and $${d\over dr} (r^\sigma K(r)) \geq 0$$, or $$1<p \leq (n+2)/(n-2)$$, and $${d\over dr} (r^{\sigma/2} K(r))\geq 0$$ and $$\not \equiv 0$$, where $$\sigma =n+2- p(n-2)$$. For the problem (*) with $$\mathbb{R}^n \setminus \{0\}$$ instead of $$\mathbb{R}^n$$, a $$C^2(\mathbb{R}^n \setminus \{0\})$$ solution is proved to be radially symmetric under the monotonicity condition of $$K(r) | r^2- \overline r^2 |^{\sigma/2}$$ in $$(0,\overline r]$$ and in $$[\overline r,\infty)$$ respectively.
The proofs are based on the Kelvin transformation and the repeated application of the moving plane method.

### MSC:

 35J60 Nonlinear elliptic equations

### Keywords:

monotonicity condition; Kelvin transformation
Full Text:

### References:

 [1] DOI: 10.1080/03605309608821182 · Zbl 0844.35025 [2] DOI: 10.1002/cpa.3160420304 · Zbl 0702.35085 [3] DOI: 10.1215/S0012-7094-91-06325-8 · Zbl 0768.35025 [4] DOI: 10.1215/S0012-7094-95-07814-4 · Zbl 0839.35014 [5] DOI: 10.1215/S0012-7094-85-05224-X · Zbl 0592.35048 [6] Hu B., Differential Integral Equations 7 pp 301– (1994) [7] DOI: 10.1080/03605309108820770 · Zbl 0741.35014 [8] Li C., Invent. Math 123 pp 221– (1996) [9] DOI: 10.1215/S0012-7094-88-05740-7 · Zbl 0674.53048 [10] DOI: 10.1007/BF00387896 · Zbl 0764.35014 [11] DOI: 10.1007/BF01053462 · Zbl 0705.35039 [12] DOI: 10.1215/S0012-7094-95-08016-8 · Zbl 0846.35050 [13] Lou Y., Differential Equations 80 (1995) [14] Kusano T., Funkcial. Ekvac 30 pp 269– (1987) [15] Terracini S., Differential Integral Equations 8 pp 1911– (1995) · Zbl 0835.35055 [16] DOI: 10.1002/cpa.3160340406 · Zbl 0465.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.